I made time to try out the Double Wrap Letter Fold with a sheet of A4. As I suspected, using this sized paper requires a little modification to make the model work. After crumpling several sheets I found a set of creases that works.
Here is the result. I didn't have any letterhead paper so I used some colored pencils to make it clear where the colored (letterhead) side or the paper shows on the outside of the folded model.
It looks a little different, both front and back, but the general idea is the same.
The down side is that there is less room on the inside to write your letter. When the paper is in the normal longitudinal orientation, the top and bottom margins are rather wide.
Here is the Crease Pattern.
The last (optional) crease is no longer required.
The method for finding the long side walls has changed. To make the model work you need a crease at the 7/32nd mark. On the right side of the pattern I have shown how this is done. Follow the numbers. To get the number 5 crease just match a long side to the "4" line on the other side of the mid-line.